Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted. Find a reasonable domain and range for the function \(C\).

Step-by-Step Solution

Answer

Domain: {2005, 2006, 2007, 2008, 2009, 2010}. Range: {11000, 11205, 11410, 11615, 11820, 12025}.

Step 1: Understand the Problem

We are given that the number of people afflicted with the common cold decreases by 205 each year starting from 12,025 in 2005, continuing until 2010. We need to find suitable domains and ranges for the function representing this relationship.

Step 2: Determine the Domain

The domain represents the range of input values, which in this case are the years from 2005 to 2010. Therefore, the domain is the set of integers from 2005 to 2010 inclusive: \( \{2005, 2006, 2007, 2008, 2009, 2010\} \).

Step 3: Calculate Afflicted People Each Year

Calculate the number of people afflicted in each year by subtracting 205 for each subsequent year starting from 12,025 in 2005: - 2005: 12,025 - 2006: 12,025 - 205 = 11,820 - 2007: 11,820 - 205 = 11,615 - 2008: 11,615 - 205 = 11,410 - 2009: 11,410 - 205 = 11,205 - 2010: 11,205 - 205 = 11,000

Step 4: Determine the Range

The range is the set of output values of afflicted people, from 2005 to 2010. Based on the previous calculations, the range of the function is \( \{11000, 11205, 11410, 11615, 11820, 12025\} \).

Step 5: State the Domain and Range

The domain of the function \( C \) is the set of years \( \{2005, 2006, 2007, 2008, 2009, 2010\} \). The range of the function is the number of afflicted people \( \{11000, 11205, 11410, 11615, 11820, 12025\} \).

Key Concepts

Function Representation

In mathematics, a function is a way to relate an input to a corresponding output. A function can be represented in various forms like an equation, a table, or a graph. In this particular scenario, we are dealing with a linear function that decreases annually. The function's equation helps in predicting the number of cold-afflicted people over time based on initial and annual decrement values.

To represent our function, we start with the initial number of people afflicted in the first year, 2005, which is 12,025. Each subsequent year, this number decreases by 205. We can denote this relationship with a function, say, \( C(x) \). Here, \( x \) refers to the year represented as a number starting from \( x = 2005 \). The function can be represented as:

• \( C(x) = 12,025 - 205(x - 2005) \)

This equation means that for each year \( x \), the number of people infected is calculated by subtracting 205 multiplied by the difference between current year \( x \) and the base year 2005 from 12,025.

Decreasing Sequence

A decreasing sequence in mathematics is a sequence where each term is less than or equal to the previous term. In simpler terms, it's a sequence that goes down over indexes.

For the problem at hand, we see the number of people afflicted by the cold each year forms a decreasing sequence, as 205 people are fewer with each passing year. This decrease is constant across all years from 2005 to 2010.

• 2005: 12,025
• 2006: 11,820
• 2007: 11,615
• 2008: 11,410
• 2009: 11,205
• 2010: 11,000

Each subsequent term in the sequence simply subtracts 205 from the previous term, forming a clear and predictable pattern. This kind of pattern allows us to easily calculate or predict future or past values, assuming the pattern does not change.

Yearly Calculation

Calculating values over a period, like from year to year, involves understanding and sometimes representing temporal changes. This is crucial in determining the domain and range of a function where time acts as a variable.

In this exercise, the yearly calculation focuses on finding the number of people afflicted with the cold each year between 2005 and 2010. Initially in 2005, 12,025 people were affected. By using a simple subtraction method every year, we find:

• 2005: 12,025
• 2006: 12,025 - 205 = 11,820
• 2007: 11,820 - 205 = 11,615
• 2008: 11,615 - 205 = 11,410
• 2009: 11,410 - 205 = 11,205
• 2010: 11,205 - 205 = 11,000

Yearly calculating helps in constructing both the domain and range. Here, the domain consists of the years {2005, 2006, 2007, 2008, 2009, 2010}, and the range is the set of afflicted amounts {11,000, 11,205, 11,410, 11,615, 11,820, 12,025}. By listing these values, we can quickly analyze trends or changes over several years.